Abstract

Tachyonic spectral densities of ultra-relativistic electron populations are fitted to the γ-ray spectra of two TeV blazars, the BL Lacertae objects 1ES 0229+200 and 1ES 0347-121. The spectral maps are compared to Galactic TeV sources, the γ-ray binary LS 5039 and the supernova remnant W28. In contrast to TeV photons, the extragalactic tachyon flux is not attenuated by interaction with the cosmic background light; there is no absorption of tachyonic γ-rays via pair creation, as tachyons do not interact with infrared background photons. The curvature of the observed γ-ray spectra is intrinsic, caused by the Boltzmann factor of the electron densities, and reproduced by a tachyonic cascade fit. In particular, the curvature in the spectral map of the Galactic microquasar is more pronounced than of the two extragalactic γ-ray sources. Estimates of the thermodynamic parameters of the thermal or, in the case of supernova remnant W28, shock-heated nonthermal electron plasma generating the tachyon flux are obtained from the spectral fits.

1. Introduction

The goal is to point out evidence for superluminal γ-rays from two active galactic nuclei, the BL Lacertae objects 1ES 0229+200, cf. Refs. [1], [2], [3] and [4], and 1ES 0347-121, cf. Refs. [5], [6] and [7]. Spectral maps of these TeV blazars have recently been obtained by means of ground-based imaging air Cherenkov detectors [4] and [7]. Here, a tachyonic cascade fit is performed to the γ-ray spectrum of these blazars. In contrast to electromagnetic γ-rays, tachyons cannot interact with infrared background photons, so that there is no attenuation of the extragalactic tachyon flux by electron–positron pair production. The observed spectrum is already the intrinsic one without any need of absorption correction as required in electromagnetic spectral fits [8] and [9]. The spectral curvature is generated by the Boltzmann factor of the thermal electron plasma in the galactic nuclei.

Tachyonic cascade spectra are obtained by averaging the superluminal spectral densities of individual electrons with ultra-relativistic Fermi distributions. We use the averaged radiation densities to perform spectral fits to the γ-ray spectra of the mentioned BL Lacertae objects as well as the microquasar LS 5039 and the supernova remnant W28. Tachyonic cascade spectra generated by thermal electron populations in the active galactic nuclei (AGNs) provide excellent fits to the observed γ-ray flux. The temperature, source count, and internal energy of the ultra-relativistic electron plasma are obtained from the spectral fits. The spectral map of the Galactic microquasar is no less curved than the cascades in the AGN spectra. Moreover, the spectral curvature of the AGNs does not increase with increasing redshift, which is further evidence for an unattenuated extragalactic γ-ray flux.

The tachyonic radiation field is a Proca field with negative mass-square,

coupled to an electron current , where m_t is the mass of the superluminal Proca field A_α, and q the tachyonic charge carried by the subluminal current [10], [11] and [12]. The mass term in (1) is added with a positive sign, and the sign convention for the metric is diag(−1,1,1,1), so that is the negative mass-square of the radiation field. The sign convention for the field tensor is F_μν=A_ν,μ−A_μ,ν. The negative mass-square refers to the radiation field rather than the current, in contrast to the traditional approach based on superluminal source particles emitting electromagnetic radiation [13] and [14]. The superluminal radiation field does not carry any kind of charge, tachyonic charge q is a property of subluminal particles, as is electric charge [15]. The field equations derived from (1) read , and can equivalently be written as Proca equation, , subject to the Lorentz condition , which follows from current conservation . An estimate of the tachyon–electron mass ratio obtained from hydrogenic Lamb shifts is m_t/m≈1/238 [16].

In Section 2, we further elaborate on the tachyonic Proca equation by comparing to electromagnetic theory, and assemble the tachyonic spectral averages employed in the fits, which are based on the transversal and longitudinal radiation densities generated by a free electronic spinor current. In Section 3, the spectral fitting is explained, and intrinsic spectral curvature is argued by comparing the cascade spectra of the above-mentioned Galactic and extragalactic TeV sources. The conclusions are summarized in Section 4.

2. Spectral maps in the γ-ray bands: Origin and structure of tachyonic cascade spectra

The Lagrangian (1) resembles its electrodynamic counterpart, but the negative mass-square of the Proca field causes striking differences. Apart from the superluminal speed of the tachyonic quanta, the radiation is partially longitudinally polarized [12], the gauge freedom is broken, and freely propagating charges can radiate superluminal quanta. The analogy to Maxwell's theory becomes even more transparent in 3D. The tachyonic E and B fields are related to the vector potential A_α=(A₀,A) by E=A₀−∂A/∂t and B=rotA, and the field equations decompose into

where we identified j^μ=(ρ,j). The vector potential is completely determined by the current and the field strengths; there is no gauge freedom owing to the tachyon mass. Another major difference to electromagnetic theory is the emergence of longitudinal wave propagation. To see this, we consider monochromatic waves, , and analogously for the scalar potential and the field strengths, with amplitudes , and , respectively. We write k=k(ω)k₀, with a unit wave vector k₀. On substituting this ansatz into the free field equations (2), we find the dispersion relation . The transversality condition on the vector potential is , so that the remaining transversal amplitudes are , and . If the product does not vanish, the modes must be longitudinal, satisfying . In this case, , and , so that a longitudinal plane wave has no magnetic component.

The spectral fits in Section 3 are based on the quantized tachyonic radiation densities

generated by the spinor current of a uniformly moving electron [11]. The superscripts T and L refer to the transversal/longitudinal polarization components defined by and Δ^L=0. γ is the electronic Lorentz factor, α_q the tachyonic fine structure constant, and m_t the tachyon mass. In the γ-ray broadband fit of supernova remnant W28, the radiation frequencies ω stretch over five decades, cf. Section 3. The units =c=1 can easily be restored. We use the Heaviside–Lorentz system, so that α_q=q²/(4πc)≈1.0×10⁻¹³ and m_t≈2.15 keV/c²; the tachyon–electron mass ratio is m_t/m≈1/238. These estimates are obtained from Lamb shifts of hydrogenic ions [16]. A spectral cutoff occurs at

Only frequencies in the range 0ωω_max(γ) can be radiated by a uniformly moving electron, the tachyonic spectral densities p^T,L(ω,γ) being cut off at the break frequency ω_max. A positive ω_max(γ) requires Lorentz factors exceeding the threshold μ_t in (4), since ω_max(μ_t)=0. The lower threshold on the speed of the electron for radiation to occur is thus υ_min=m_t/(2mμ_t). The tachyon–electron mass ratio gives υ_min/c≈2.1×10⁻³, which is roughly the speed of the Galaxy in the microwave background [17].

The radiation densities (3) refer to a single spinning charge with Lorentz factor γ; we average them with a Fermi power-law distribution [18] and [19],

The particle number is found as , where γ₁ is the lower edge of Lorentz factors in the source population. m is the electron mass, and the exponential cutoff is related to the electron temperature by β=m/(kT). The power-law exponent δ ranges in a narrow interval in astrophysical spectral averages [20], [21], [22], [23], [24], [25], [26], [27] and [28], which are usually done in the classical limit (13) with 0δ4. Otherwise, there are no conceptual constraints on the power-law index of these stationary non-equilibrium distributions. Thermal equilibrium is recovered with δ=0 and γ₁=1; the spectral fits of the blazars and the microquasar in Fig. 1, Fig. 2 and Fig. 3 are performed with equilibrium distributions. The shocked electron plasma of supernova remnant W28 requires δ≈3.6 to fit the steep power-law slope in Fig. 4. The exponent in (5) defines the fugacity ; we use a hat to distinguish from the electron index customarily defined as α=δ−2, cf. (13). The fugacity is related to the chemical potential μ by .

The spectral average of the radiation densities (3) is carried out as

where θ is the Heaviside step function. The spectral range of densities (3) is bounded by ω_max in (4), so that the solution of defines the minimal electronic Lorentz factor for radiation at this frequency [29],

The average (6) can be reduced to the fermionic spectral functions

with lower integration boundary γ₁μ_t, cf. after (4). This threshold Lorentz factor γ₁ defines the break frequency,

which separates the spectrum into a low- and high-frequency band [30]. By making use of the spectral functions (8), we can write the averaged radiation densities (6) as

with in (7) and ω₁ in (9), so that . The superscripts T and L denote the transversal and longitudinal radiation components, cf. (3). The spectral functions F^T,L(ω,γ₁) in (10) are obtained by substituting radiation densities (3) into the integral representation (8),

where the weight factors f_k read

with density dρ_F(γ) in (5).

The quasiclassical fugacity expansion of the spectral functions (11) is found by expanding density (5) in ascending powers of . In leading order, dρ_F(γ)dρ_α,β(γ),

This power-law density is the classical limit of the fermionic density dρ_F. As mentioned above, exponent in the normalization factor A_α,β defines the fugacity, and is not to be confused with the electron index α=δ−2 in (13). The leading order of the reduced spectral functions (12) can be written as incomplete Γ-function, f_k(γ₁)A_α,ββ^δ−kΓ(k−δ,βγ₁), obtained by replacing the fermionic dρ_F in the weights (12) by the Boltzmann power-law density (13).

The classical limit of the fermionic spectral functions F^T,L(ω,γ₁) in (8) is the Boltzmann average [31]

Here, is the classical tachyonic spectral density, recovered by dropping all terms containing m_t/m ratios in (3) and (4). In particular, and , cf. (3). The classical spectral functions B^T,L(ω,γ₁) are obtained from the Fermi functions F^T,L(ω,γ₁) in (11) by dropping the m_t/m terms, and substituting the leading order of the fugacity expansion of the weights f_k(γ₁) as stated after (13). The classical limit of the fermionic spectral average p^T,L(ω)_F in (10) thus reads

where is the classical limit of the break frequency (9).

3. Spectral curvature of Galactic and extragalactic TeV γ-ray sources: Does distance matter?

The spectral fits of the active galactic nuclei (AGNs) and the Galactic TeV sources in Fig. 1, Fig. 2, Fig. 3 and Fig. 4 are based on the E²-rescaled flux densities

where d is the distance to the source and p^T,L(ω)_α,β the spectral average (15) (with ω=E/). The fits are done with the unpolarized flux density dN^T+L=dN^T+dN^L of two electron populations ρ_i=1,2, cf. Table 1. Each electron density generates a cascade ρ_i, and the wideband comprises two cascade spectra labeled ρ₁ and ρ₂ in the figures. As for the electron count, , we use a rescaled parameter for the fit,

which is independent of the distance estimate in (16). Here, implies the tachyon mass in keV units, that is, we put m_t≈2.15 in the spectral densities (3). At γ-ray energies, only a tiny α_q/α_e-fraction (the ratio of tachyonic and electric fine structure constants) of the tachyon flux is absorbed by the detector, which requires a rescaling of the electron count n₁, so that the actual number of radiating electrons is , cf. Ref. [12]. We thus find the electron count as , where defines the tachyonic flux amplitude extracted from the spectral fit. (In Table 1, the subscript 1 of and has been dropped.) Electron temperature and cutoff parameter in the Boltzmann factor of density (13) are related by kT[TeV]≈5.11×10⁻⁷/β, and the energy estimates of the thermal cascades in Table 1 are based on , cf. Ref. [18]. (The renormalized count is to be identified with the particle number N in the thermodynamic functions discussed in this reference.) The distance estimates of the AGNs are based on dcz/H₀, with the Hubble distance c/H₀≈4.4×10³ Mpc (that is, h₀≈0.68). Hence, d[Mpc]≈4.4×10³z, and , cf. Table 1.

Fig. 1 shows the tachyonic spectral map of the BL Lacertae object (BL Lac) 1ES 0229+200, located at a redshift of z≈0.140, cf. Refs. [1], [2], [3] and [4]. The flux points were obtained with the HESS array of atmospheric Cherenkov telescopes in the Khomas Highland of Namibia [4]. The χ²-fit is done with the unpolarized tachyon flux T+L, and subsequently split into transversal and longitudinal components. The differential flux is rescaled with E² for better visibility of the spectral curvature. Temperature and source count of the electron populations generating the cascades are recorded in Table 1. In Fig. 2, we show the spectral map of the BL Lac 1ES 0347-121, at a redshift of z≈0.188, cf. Refs. [5], [6] and [7]. TeV γ-ray spectra of BL Lacs are usually assumed to be generated by inverse Compton scattering or proton–proton scattering followed by pion decay [8]. Both mechanisms result in a flux of TeV photons, assumed to be partially absorbed by interaction with infrared background photons owing to pair creation, so that the intrinsic spectrum has to be reconstructed on the basis of intergalactic absorption models depending on vaguely known cosmological input parameters [9]. The extragalactic tachyon flux is not attenuated by interaction with the background light, there is no absorption of tachyonic γ-rays. The superluminal Proca field (1) is minimally coupled to the electron current, and does not directly interact with electromagnetic radiation.

The spectral curvature apparent in double-logarithmic plots of the E²-rescaled flux densities (16) is intrinsic, caused by the Boltzmann factor of the electron populations generating the tachyon flux. The curvature present in TeV γ-ray spectra does not increase with distance, at least there is no evidence to that effect if we compare the spectral slopes of the blazars in Fig. 1 and Fig. 2, and the spectral maps of other flaring AGNs such as H1426+428 at z≈0.129 and 1ES 1959+650 at z≈0.047, cf. Ref. [11]. Further evidence for intrinsic spectral curvature is provided by Fig. 3, depicting the spectral map of the microquasar LS 5039, a compact object orbiting a massive O-star [32], [33] and [34]. The spectral curvature of this Galactic binary is even more pronounced than of the BL Lacs in Fig. 1 and Fig. 2. The same holds true for the HESS spectral map of LS 5039 at the inferior conjunction studied in Ref. [18].

Fig. 4 depicts the γ-ray wideband of the TeV source HESS J1801-233 and the coincident EGRET point source 3EG J1800-2338 [35], [36] and [37]. The extended TeV source is located on the northeastern rim of the supernova remnant (SNR) W28, a mixed-morphology SNR interacting with molecular clouds [38] and [39]. The spectral map of SNR W28 in Fig. 4 is to be compared to the unpulsed γ-ray spectrum of the Crab Nebula, cf. Fig. 1 in Ref. [11], the spectral map of SNR RX J1713.7-3946 in Fig. 2 of Ref. [11], and in particular to the unidentified TeV source TeV J2032+4130 in conjunction with the associated EGRET source 3EG J2033+4118, cf. Fig. 6 of Ref. [12]. A spectral plateau in the MeV to GeV range occurs frequently in spectral maps of TeV γ-ray sources, and can easily be fitted with tachyonic cascade spectra, in contrast to electromagnetic inverse-Compton fits. SNR W28 is located at a distance of 1.9 kpc [38]. The thermal cascade in the MeV range (ρ₂ in Fig. 4) preceding the spectral plateau is strongly curved; the power-law slope of the ρ₁ cascade also terminates in exponential decay, but outside the presently accessible TeV range shown in the figure. The cutoff temperature of the nonthermal electron density generating cascade ρ₁ is too high to bend the power-law slope in the TeV range covered in Fig. 4, so that the internal energy of the shock-heated plasma could not be determined from the χ²-fit, in contrast with the power-law index α and the threshold Lorentz factor γ₁, cf. caption to Table 1.

4. Conclusion

The spectral maps of two TeV γ-ray blazars have been fitted with tachyonic cascade spectra and compared to a Galactic γ-ray binary and supernova remnant. Table 1 contains estimates of the thermodynamic parameters of the electron populations generating the superluminal cascades. The spectral curvature is intrinsic and reproduced by the tachyonic spectral densities (3) averaged with ultra-relativistic thermal electron distributions, cf. Fig. 1, Fig. 2 and Fig. 3. The shocked electron plasma in SNR W28 requires a nonthermal power-law distribution to adequately reproduce the TeV cascade in Fig. 4. The curvature in the γ-ray spectra of BL Lacs is uncorrelated with distance, so that absorption of electromagnetic radiation due to interaction with infrared photons is not an attractive explanation of spectral curvature. By contrast, there is no attenuation of the extragalactic tachyon flux, as tachyons cannot interact with cosmic background photons, so that the observed cascades are the intrinsic spectrum. The distance of the AGNs does not show in the spectral curvature, as tachyonic cascades are unaffected by background photons.

Acknowledgements

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.